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Theorem asclfval 20570
 Description: Function value of the algebraic scalars function. (Contributed by Mario Carneiro, 8-Mar-2015.)
Hypotheses
Ref Expression
asclfval.a 𝐴 = (algSc‘𝑊)
asclfval.f 𝐹 = (Scalar‘𝑊)
asclfval.k 𝐾 = (Base‘𝐹)
asclfval.s · = ( ·𝑠𝑊)
asclfval.o 1 = (1r𝑊)
Assertion
Ref Expression
asclfval 𝐴 = (𝑥𝐾 ↦ (𝑥 · 1 ))
Distinct variable groups:   𝑥,𝐾   𝑥, 1   𝑥, ·   𝑥,𝑊
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem asclfval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 asclfval.a . 2 𝐴 = (algSc‘𝑊)
2 fveq2 6646 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3 asclfval.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
42, 3eqtr4di 2851 . . . . . . 7 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
54fveq2d 6650 . . . . . 6 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
6 asclfval.k . . . . . 6 𝐾 = (Base‘𝐹)
75, 6eqtr4di 2851 . . . . 5 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
8 fveq2 6646 . . . . . . 7 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
9 asclfval.s . . . . . . 7 · = ( ·𝑠𝑊)
108, 9eqtr4di 2851 . . . . . 6 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
11 eqidd 2799 . . . . . 6 (𝑤 = 𝑊𝑥 = 𝑥)
12 fveq2 6646 . . . . . . 7 (𝑤 = 𝑊 → (1r𝑤) = (1r𝑊))
13 asclfval.o . . . . . . 7 1 = (1r𝑊)
1412, 13eqtr4di 2851 . . . . . 6 (𝑤 = 𝑊 → (1r𝑤) = 1 )
1510, 11, 14oveq123d 7157 . . . . 5 (𝑤 = 𝑊 → (𝑥( ·𝑠𝑤)(1r𝑤)) = (𝑥 · 1 ))
167, 15mpteq12dv 5116 . . . 4 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠𝑤)(1r𝑤))) = (𝑥𝐾 ↦ (𝑥 · 1 )))
17 df-ascl 20549 . . . 4 algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠𝑤)(1r𝑤))))
1816, 17, 6mptfvmpt 6969 . . 3 (𝑊 ∈ V → (algSc‘𝑊) = (𝑥𝐾 ↦ (𝑥 · 1 )))
19 fvprc 6639 . . . . 5 𝑊 ∈ V → (algSc‘𝑊) = ∅)
20 mpt0 6463 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑥 · 1 )) = ∅
2119, 20eqtr4di 2851 . . . 4 𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥 · 1 )))
22 fvprc 6639 . . . . . . . . 9 𝑊 ∈ V → (Scalar‘𝑊) = ∅)
233, 22syl5eq 2845 . . . . . . . 8 𝑊 ∈ V → 𝐹 = ∅)
2423fveq2d 6650 . . . . . . 7 𝑊 ∈ V → (Base‘𝐹) = (Base‘∅))
25 base0 16531 . . . . . . 7 ∅ = (Base‘∅)
2624, 25eqtr4di 2851 . . . . . 6 𝑊 ∈ V → (Base‘𝐹) = ∅)
276, 26syl5eq 2845 . . . . 5 𝑊 ∈ V → 𝐾 = ∅)
2827mpteq1d 5120 . . . 4 𝑊 ∈ V → (𝑥𝐾 ↦ (𝑥 · 1 )) = (𝑥 ∈ ∅ ↦ (𝑥 · 1 )))
2921, 28eqtr4d 2836 . . 3 𝑊 ∈ V → (algSc‘𝑊) = (𝑥𝐾 ↦ (𝑥 · 1 )))
3018, 29pm2.61i 185 . 2 (algSc‘𝑊) = (𝑥𝐾 ↦ (𝑥 · 1 ))
311, 30eqtri 2821 1 𝐴 = (𝑥𝐾 ↦ (𝑥 · 1 ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ∅c0 4243   ↦ cmpt 5111  ‘cfv 6325  (class class class)co 7136  Basecbs 16478  Scalarcsca 16563   ·𝑠 cvsca 16564  1rcur 19248  algSccascl 20546 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-slot 16482  df-base 16484  df-ascl 20549 This theorem is referenced by:  asclval  20571  asclfn  20572  asclf  20573  rnascl  20583  ressascl  20588  asclpropd  20589  rnasclg  39451
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